Numerical analysis of the representer method applied to reservoir modeling
The representer method is a data assimilation scheme that has received attention in the traditionally “data-rich” fields of oceanography and meteorology. Now, advances in instrumenting and imaging subsurface reservoirs are yielding large data sets, making data assimilation vital to petroleum and environmental engineering. However, some numerical properties of the representer method have not been fully explored. Thus, there are great opportunities to apply and study the methodology in these fields. The theme of this dissertation is analyzing the numerical properties of the representer method, particularly as applied to subsurface flow simulations. When modeling complex systems, such as oil reservoirs, measured data from the system often are not matched by the model even after consideration of measurement and computational errors. Because the mathematical model is the best representation of the system away from the measurements, data assimilation arises naturally as a way to “smooth” data, using the model, to obtain a better representation of the system. A development of a new framework to derive error estimates begins the study of the numerical properties of the representer method. In particular, an a priori estimate for mixed finite elements applied to single-phase Darcy flow in porous media is derived. This estimate demonstrates that the rate of convergence of the numerical method is maintained through the implementation of the representer method. The framework is also used to prove a posteriori error estimates, which are key ingredients in adaptive grid refinement strategies. Since it is possible to use different grids within the representer algorithm, adaptive refinement of grids is studied with the goal of improving the accuracy and computational efficiency of the implementation. Furthermore, the representer method applied to the oil/water model for reservoirs, a nonlinear model, is derived and numerically implemented for the first time. A key concern is the linearization of the Euler-Lagrange equations that converges appropriately; otherwise, nonphysical solutions are introduced. The effects of linearization and the computational costs of the representer method are compared with the ensemble Kalman filter.